## ABSTRACT

This thesis presents an overview on the theory of stopping times,

martingales and Brownian motion which are the foundations of

stochastic modeling. We started with a detailed study of discrete

stopping times and their properties. Next, we reviewed

the theory of martingales and saw an application to solving the

problem of “extinction of populations”. After that, we studied

stopping times in the continuous case and finally, we treated

extensively the concepts of Brownian motion and the Wienner integral.

KeyWords. Stochastic Processes, Stopping times, Martingales,

Galton-Watson branching process, Brownian motion.

** **

## TABLE OF CONTENTS

1. Introduction 23

2. Conditional Expectation 25

3. Definitions and Basic Properties 28

4. Maximal Inequalities 33

5. Almost sure convergence of Super or Sub-Martingale

and Krickeberg Decomposition 38

6. L1 convergence and Regular Martingales 42

7. Doob’s Decomposition for a submartingale 49

Chapter 4. Watson-Galton Stochastic process : Extinction

of populations 51

1. Introduction 51

2. Martingale Approach 52

3. Extinction Probability Approach 56

Part 2. Continuous Stochastic Modeling 65

Chapter 5. Stopping Time and Measurable Stochastic

Processes 67

1. Stopped Stochastic processes in the continuous

case 67

Chapter 6. Introduction to the Brownian Motion 73

1. Kolmogorov Construction of the Brownian Motion 73

2. Characterizations and Tranformations of the

Brownian Motion 76

3. Tranformations 78

4. Standard Brownian Motion 80

CONTENTS iii

5. Elements of random Analysis using the standard

Brownian motion 92

Chapter 7. Poisson Stochastic Processes 111

1. Description by exponential inter-arrival 111

2. Counting function 115

3. Approach of the Kolmogorov Existence Theorem 121

4. More properties for the Standard Poisson Process 124

5. Kolmogorov equations 138

Part 3. Stochastic Integration 147

Chapter 8. Itˆo Integration or Stochastic Calculus 149

1. Regularity of paths of stochastic processes 150

2. Definition and justification of the Itˆo Stochastic

integrals 153

3. The Itˆo Integral 166

4. Computations 167

Conclusions and Perspectives 175

5. Achievements 175

6. Perspectives 176

Bibliography 177

## CHAPTER ONE

General Introduction

1. The context

The present dissertation should be placed in the project to build

within the African University of Sciences and Technologies a

research team in Stochastics and Statistics.

For a significant number of years, the course Measure Theory

and Integration (MTI) is taught. In the two precedent Master

classes, the course (MTI) has been extensively developed. The

time allocated to this course allows now to cover the contents

of the main reference of the course which is the exposition of

Lo (2018).

That content exposed in seven hundred pages is intended to allow

the reader to train himself on the knowledge broken into

exercises.

This full course of (MTI) should be the basis of two teams of

research in AUST:

(A) a team of research in Abstract integration and in Set-valued

Integrations.

1

2 1. GENERAL INTRODUCTION

(B) a team on Stochastics and applications in Finance, Biology,

Genetics, Population, etc.

The basis in Probability theory which is beneath (B) will lead

to a branch of research in :

(C) Statistical Methods and Applied Statistics.

In setting up the described process, in its Probability theory

component, the first step consisted in the development of

the course of Foundation of Probability Theory (MFPT) (Lo (2018)).

This book was exposed in 2019 as a PhD course in AUST.

The aim of this dissertation is to gather the mathematical tools

for stochastic modeling, or at least to gather a great deal of

them in a consistent text based on the books of (MTI) and (MFPT).

So, the dissertation will open the doors of first thesis in Stochastics

in AUST or will serve the future candidates for theses in

Stochastics In AUST.

2. Stochastic Modeling

In real, many phenomena are described by sequence of random variables

or family of random variables. Those described by a sequence

require discrete stochastic modeling while those described

2. STOCHASTIC MODELING 3

by an arbitrary family requires continuous stochastic modeling.

For example :

(a) In gambling, the surplus of a gambler at a discrete time

n is a random variable Xn. Here one may be interested in the

possibility of the gambler losing all of his money and to get

ruined.

(b) Let us assume that some population begins with a patriarch

which reproduces a random number offspring at time n = 1. At

any time n+1, each of the offspring reproduced at time n gives

a random number of offspring. So the total number of new members

at time n is a random number Xn. A natural question is

: is there any possibility that the population comes to extinction,

that is no offspring are made at some time N. We might

also want to have an estimation of the number of offspring for

large values of n, whether Xn becomes stable or increases to

infinity (case of China in the past) or decreases to zero (actual

situation in some European countries).

In these two cases, we face discrete stochastic modeling.

(c) Let us suppose that an insurance company has a surplus St

at time t. It continues collecting premiums from clients with

Pt the total of premium collected at time t, the return of its

investments of the premium with Ct the total investments returns

at time t and paying the claims to clients with Lt the total amount

4 1. GENERAL INTRODUCTION

of losses payed to clients. The surplus of the company at time

t is

St = u + Pt + Ct ? Lt;

where u is the initial surplus at tome t = 0 or capital. The

worse event the company wants to avoid is the ruin situation

at time a t0, which is the first time where St 0.

Dealing with Situation (c) is done through continuous time stochastic

modeling.

In this dissertation, we will provide interesting parts of the

theory beneath such stochastic modeling.

3. Scope of the dissertation

We divide the dissertation into three parts.

./ The first part deals with discrete stochastic modeling. We

will introduce two very important notions, that is, the notion

of stopping times and theory of martingales.

As a first example, we study the extinction question of a sequence

of a population, as described in Situation (b) above in

specific conditions.

3. SCOPE OF THE DISSERTATION 5

./ The second part deals with continuous stochastic modeling.

Here again, We will introduce to continuous versions for stopping

times and most importantly, we are going to complete this

section with an introduction to Brownian Motion and present a

thorough study of it.

./ The third part is an opening to Stochastic Integration and

Stochastic Differential equations.

Generally, the contents I summarized here can be found in the

most important books of the discipline. However, I particularly

used Lo`eve (1997), Chung (1974), Neveu (1965) and Lo

(2018) for the fundamental modern probability theory, Neveu

(1975) for discrete martingale, Billingsley (1995), Taylor and

Karlin (1987) for the introduction to stochastic processes and

Kuo (2000) for the stochastic calculus. Gathering all this

materials and using them in a coherent way was possible in the

frame of the series on probability and statistics in which Professor

Lo introduces to the most inner secret of those disciplines

in a series of books (Lo (2018), Lo (2018), Lo (2019),

etc.) I am grateful to be able to benefit from that frame that

helped me to reach so many things in a few months.

I am aware that reading and mastering the the key elements of

Stochastics and trying to realize the described content is a

very difficult and heavy challenge. But with the help of the

6 1. GENERAL INTRODUCTION

leaders of AUST, especially the HOD of Pure and Applied Mathematics,

with the full supervision of professor Gane Samb Lo,

we humbly think that we had a firm introduction to stochastic

modeling and we are ready to go further to research

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